3.417 \(\int \frac{A+B x}{x^{5/2} (a+c x^2)} \, dx\)

Optimal. Leaf size=278 \[ -\frac{\sqrt [4]{c} \left (\sqrt{a} B-A \sqrt{c}\right ) \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} \sqrt{x}+\sqrt{a}+\sqrt{c} x\right )}{2 \sqrt{2} a^{7/4}}+\frac{\sqrt [4]{c} \left (\sqrt{a} B-A \sqrt{c}\right ) \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} \sqrt{x}+\sqrt{a}+\sqrt{c} x\right )}{2 \sqrt{2} a^{7/4}}+\frac{\sqrt [4]{c} \left (\sqrt{a} B+A \sqrt{c}\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}}\right )}{\sqrt{2} a^{7/4}}-\frac{\sqrt [4]{c} \left (\sqrt{a} B+A \sqrt{c}\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}}+1\right )}{\sqrt{2} a^{7/4}}-\frac{2 A}{3 a x^{3/2}}-\frac{2 B}{a \sqrt{x}} \]

[Out]

(-2*A)/(3*a*x^(3/2)) - (2*B)/(a*Sqrt[x]) + ((Sqrt[a]*B + A*Sqrt[c])*c^(1/4)*ArcTan[1 - (Sqrt[2]*c^(1/4)*Sqrt[x
])/a^(1/4)])/(Sqrt[2]*a^(7/4)) - ((Sqrt[a]*B + A*Sqrt[c])*c^(1/4)*ArcTan[1 + (Sqrt[2]*c^(1/4)*Sqrt[x])/a^(1/4)
])/(Sqrt[2]*a^(7/4)) - ((Sqrt[a]*B - A*Sqrt[c])*c^(1/4)*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*c^(1/4)*Sqrt[x] + Sqrt[c
]*x])/(2*Sqrt[2]*a^(7/4)) + ((Sqrt[a]*B - A*Sqrt[c])*c^(1/4)*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*c^(1/4)*Sqrt[x] + S
qrt[c]*x])/(2*Sqrt[2]*a^(7/4))

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Rubi [A]  time = 0.272735, antiderivative size = 278, normalized size of antiderivative = 1., number of steps used = 12, number of rules used = 8, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.4, Rules used = {829, 827, 1168, 1162, 617, 204, 1165, 628} \[ -\frac{\sqrt [4]{c} \left (\sqrt{a} B-A \sqrt{c}\right ) \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} \sqrt{x}+\sqrt{a}+\sqrt{c} x\right )}{2 \sqrt{2} a^{7/4}}+\frac{\sqrt [4]{c} \left (\sqrt{a} B-A \sqrt{c}\right ) \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} \sqrt{x}+\sqrt{a}+\sqrt{c} x\right )}{2 \sqrt{2} a^{7/4}}+\frac{\sqrt [4]{c} \left (\sqrt{a} B+A \sqrt{c}\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}}\right )}{\sqrt{2} a^{7/4}}-\frac{\sqrt [4]{c} \left (\sqrt{a} B+A \sqrt{c}\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}}+1\right )}{\sqrt{2} a^{7/4}}-\frac{2 A}{3 a x^{3/2}}-\frac{2 B}{a \sqrt{x}} \]

Antiderivative was successfully verified.

[In]

Int[(A + B*x)/(x^(5/2)*(a + c*x^2)),x]

[Out]

(-2*A)/(3*a*x^(3/2)) - (2*B)/(a*Sqrt[x]) + ((Sqrt[a]*B + A*Sqrt[c])*c^(1/4)*ArcTan[1 - (Sqrt[2]*c^(1/4)*Sqrt[x
])/a^(1/4)])/(Sqrt[2]*a^(7/4)) - ((Sqrt[a]*B + A*Sqrt[c])*c^(1/4)*ArcTan[1 + (Sqrt[2]*c^(1/4)*Sqrt[x])/a^(1/4)
])/(Sqrt[2]*a^(7/4)) - ((Sqrt[a]*B - A*Sqrt[c])*c^(1/4)*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*c^(1/4)*Sqrt[x] + Sqrt[c
]*x])/(2*Sqrt[2]*a^(7/4)) + ((Sqrt[a]*B - A*Sqrt[c])*c^(1/4)*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*c^(1/4)*Sqrt[x] + S
qrt[c]*x])/(2*Sqrt[2]*a^(7/4))

Rule 829

Int[(((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_)))/((a_) + (c_.)*(x_)^2), x_Symbol] :> Simp[((e*f - d*g)*(d
+ e*x)^(m + 1))/((m + 1)*(c*d^2 + a*e^2)), x] + Dist[1/(c*d^2 + a*e^2), Int[((d + e*x)^(m + 1)*Simp[c*d*f + a*
e*g - c*(e*f - d*g)*x, x])/(a + c*x^2), x], x] /; FreeQ[{a, c, d, e, f, g, m}, x] && NeQ[c*d^2 + a*e^2, 0] &&
FractionQ[m] && LtQ[m, -1]

Rule 827

Int[((f_.) + (g_.)*(x_))/(Sqrt[(d_.) + (e_.)*(x_)]*((a_) + (c_.)*(x_)^2)), x_Symbol] :> Dist[2, Subst[Int[(e*f
- d*g + g*x^2)/(c*d^2 + a*e^2 - 2*c*d*x^2 + c*x^4), x], x, Sqrt[d + e*x]], x] /; FreeQ[{a, c, d, e, f, g}, x]
&& NeQ[c*d^2 + a*e^2, 0]

Rule 1168

Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[a*c, 2]}, Dist[(d*q + a*e)/(2*a*c),
Int[(q + c*x^2)/(a + c*x^4), x], x] + Dist[(d*q - a*e)/(2*a*c), Int[(q - c*x^2)/(a + c*x^4), x], x]] /; FreeQ
[{a, c, d, e}, x] && NeQ[c*d^2 + a*e^2, 0] && NeQ[c*d^2 - a*e^2, 0] && NegQ[-(a*c)]

Rule 1162

Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[(2*d)/e, 2]}, Dist[e/(2*c), Int[1/S
imp[d/e + q*x + x^2, x], x], x] + Dist[e/(2*c), Int[1/Simp[d/e - q*x + x^2, x], x], x]] /; FreeQ[{a, c, d, e},
x] && EqQ[c*d^2 - a*e^2, 0] && PosQ[d*e]

Rule 617

Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*Simplify[(a*c)/b^2]}, Dist[-2/b, Sub
st[Int[1/(q - x^2), x], x, 1 + (2*c*x)/b], x] /; RationalQ[q] && (EqQ[q^2, 1] ||  !RationalQ[b^2 - 4*a*c])] /;
FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 204

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> -Simp[ArcTan[(Rt[-b, 2]*x)/Rt[-a, 2]]/(Rt[-a, 2]*Rt[-b, 2]), x] /
; FreeQ[{a, b}, x] && PosQ[a/b] && (LtQ[a, 0] || LtQ[b, 0])

Rule 1165

Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[(-2*d)/e, 2]}, Dist[e/(2*c*q), Int[
(q - 2*x)/Simp[d/e + q*x - x^2, x], x], x] + Dist[e/(2*c*q), Int[(q + 2*x)/Simp[d/e - q*x - x^2, x], x], x]] /
; FreeQ[{a, c, d, e}, x] && EqQ[c*d^2 - a*e^2, 0] && NegQ[d*e]

Rule 628

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[(d*Log[RemoveContent[a + b*x +
c*x^2, x]])/b, x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]

Rubi steps

\begin{align*} \int \frac{A+B x}{x^{5/2} \left (a+c x^2\right )} \, dx &=-\frac{2 A}{3 a x^{3/2}}+\frac{\int \frac{a B-A c x}{x^{3/2} \left (a+c x^2\right )} \, dx}{a}\\ &=-\frac{2 A}{3 a x^{3/2}}-\frac{2 B}{a \sqrt{x}}+\frac{\int \frac{-a A c-a B c x}{\sqrt{x} \left (a+c x^2\right )} \, dx}{a^2}\\ &=-\frac{2 A}{3 a x^{3/2}}-\frac{2 B}{a \sqrt{x}}+\frac{2 \operatorname{Subst}\left (\int \frac{-a A c-a B c x^2}{a+c x^4} \, dx,x,\sqrt{x}\right )}{a^2}\\ &=-\frac{2 A}{3 a x^{3/2}}-\frac{2 B}{a \sqrt{x}}+\frac{\left (\sqrt{a} B-A \sqrt{c}\right ) \operatorname{Subst}\left (\int \frac{\sqrt{a} \sqrt{c}-c x^2}{a+c x^4} \, dx,x,\sqrt{x}\right )}{a^{3/2}}-\frac{\left (\sqrt{a} B+A \sqrt{c}\right ) \operatorname{Subst}\left (\int \frac{\sqrt{a} \sqrt{c}+c x^2}{a+c x^4} \, dx,x,\sqrt{x}\right )}{a^{3/2}}\\ &=-\frac{2 A}{3 a x^{3/2}}-\frac{2 B}{a \sqrt{x}}-\frac{\left (\sqrt{a} B+A \sqrt{c}\right ) \operatorname{Subst}\left (\int \frac{1}{\frac{\sqrt{a}}{\sqrt{c}}-\frac{\sqrt{2} \sqrt [4]{a} x}{\sqrt [4]{c}}+x^2} \, dx,x,\sqrt{x}\right )}{2 a^{3/2}}-\frac{\left (\sqrt{a} B+A \sqrt{c}\right ) \operatorname{Subst}\left (\int \frac{1}{\frac{\sqrt{a}}{\sqrt{c}}+\frac{\sqrt{2} \sqrt [4]{a} x}{\sqrt [4]{c}}+x^2} \, dx,x,\sqrt{x}\right )}{2 a^{3/2}}-\frac{\left (\left (\sqrt{a} B-A \sqrt{c}\right ) \sqrt [4]{c}\right ) \operatorname{Subst}\left (\int \frac{\frac{\sqrt{2} \sqrt [4]{a}}{\sqrt [4]{c}}+2 x}{-\frac{\sqrt{a}}{\sqrt{c}}-\frac{\sqrt{2} \sqrt [4]{a} x}{\sqrt [4]{c}}-x^2} \, dx,x,\sqrt{x}\right )}{2 \sqrt{2} a^{7/4}}-\frac{\left (\left (\sqrt{a} B-A \sqrt{c}\right ) \sqrt [4]{c}\right ) \operatorname{Subst}\left (\int \frac{\frac{\sqrt{2} \sqrt [4]{a}}{\sqrt [4]{c}}-2 x}{-\frac{\sqrt{a}}{\sqrt{c}}+\frac{\sqrt{2} \sqrt [4]{a} x}{\sqrt [4]{c}}-x^2} \, dx,x,\sqrt{x}\right )}{2 \sqrt{2} a^{7/4}}\\ &=-\frac{2 A}{3 a x^{3/2}}-\frac{2 B}{a \sqrt{x}}-\frac{\left (\sqrt{a} B-A \sqrt{c}\right ) \sqrt [4]{c} \log \left (\sqrt{a}-\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} \sqrt{x}+\sqrt{c} x\right )}{2 \sqrt{2} a^{7/4}}+\frac{\left (\sqrt{a} B-A \sqrt{c}\right ) \sqrt [4]{c} \log \left (\sqrt{a}+\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} \sqrt{x}+\sqrt{c} x\right )}{2 \sqrt{2} a^{7/4}}-\frac{\left (\left (\sqrt{a} B+A \sqrt{c}\right ) \sqrt [4]{c}\right ) \operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1-\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}}\right )}{\sqrt{2} a^{7/4}}+\frac{\left (\left (\sqrt{a} B+A \sqrt{c}\right ) \sqrt [4]{c}\right ) \operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1+\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}}\right )}{\sqrt{2} a^{7/4}}\\ &=-\frac{2 A}{3 a x^{3/2}}-\frac{2 B}{a \sqrt{x}}+\frac{\left (\sqrt{a} B+A \sqrt{c}\right ) \sqrt [4]{c} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}}\right )}{\sqrt{2} a^{7/4}}-\frac{\left (\sqrt{a} B+A \sqrt{c}\right ) \sqrt [4]{c} \tan ^{-1}\left (1+\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}}\right )}{\sqrt{2} a^{7/4}}-\frac{\left (\sqrt{a} B-A \sqrt{c}\right ) \sqrt [4]{c} \log \left (\sqrt{a}-\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} \sqrt{x}+\sqrt{c} x\right )}{2 \sqrt{2} a^{7/4}}+\frac{\left (\sqrt{a} B-A \sqrt{c}\right ) \sqrt [4]{c} \log \left (\sqrt{a}+\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} \sqrt{x}+\sqrt{c} x\right )}{2 \sqrt{2} a^{7/4}}\\ \end{align*}

Mathematica [C]  time = 0.0143873, size = 53, normalized size = 0.19 \[ -\frac{2 \left (A \, _2F_1\left (-\frac{3}{4},1;\frac{1}{4};-\frac{c x^2}{a}\right )+3 B x \, _2F_1\left (-\frac{1}{4},1;\frac{3}{4};-\frac{c x^2}{a}\right )\right )}{3 a x^{3/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[(A + B*x)/(x^(5/2)*(a + c*x^2)),x]

[Out]

(-2*(A*Hypergeometric2F1[-3/4, 1, 1/4, -((c*x^2)/a)] + 3*B*x*Hypergeometric2F1[-1/4, 1, 3/4, -((c*x^2)/a)]))/(
3*a*x^(3/2))

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Maple [A]  time = 0.011, size = 289, normalized size = 1. \begin{align*} -{\frac{2\,A}{3\,a}{x}^{-{\frac{3}{2}}}}-2\,{\frac{B}{a\sqrt{x}}}-{\frac{Ac\sqrt{2}}{4\,{a}^{2}}\sqrt [4]{{\frac{a}{c}}}\ln \left ({ \left ( x+\sqrt [4]{{\frac{a}{c}}}\sqrt{x}\sqrt{2}+\sqrt{{\frac{a}{c}}} \right ) \left ( x-\sqrt [4]{{\frac{a}{c}}}\sqrt{x}\sqrt{2}+\sqrt{{\frac{a}{c}}} \right ) ^{-1}} \right ) }-{\frac{Ac\sqrt{2}}{2\,{a}^{2}}\sqrt [4]{{\frac{a}{c}}}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{a}{c}}}}}}+1 \right ) }-{\frac{Ac\sqrt{2}}{2\,{a}^{2}}\sqrt [4]{{\frac{a}{c}}}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{a}{c}}}}}}-1 \right ) }-{\frac{B\sqrt{2}}{4\,a}\ln \left ({ \left ( x-\sqrt [4]{{\frac{a}{c}}}\sqrt{x}\sqrt{2}+\sqrt{{\frac{a}{c}}} \right ) \left ( x+\sqrt [4]{{\frac{a}{c}}}\sqrt{x}\sqrt{2}+\sqrt{{\frac{a}{c}}} \right ) ^{-1}} \right ){\frac{1}{\sqrt [4]{{\frac{a}{c}}}}}}-{\frac{B\sqrt{2}}{2\,a}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{a}{c}}}}}}+1 \right ){\frac{1}{\sqrt [4]{{\frac{a}{c}}}}}}-{\frac{B\sqrt{2}}{2\,a}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{a}{c}}}}}}-1 \right ){\frac{1}{\sqrt [4]{{\frac{a}{c}}}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((B*x+A)/x^(5/2)/(c*x^2+a),x)

[Out]

-2/3*A/a/x^(3/2)-2*B/a/x^(1/2)-1/4/a^2*c*A*(a/c)^(1/4)*2^(1/2)*ln((x+(a/c)^(1/4)*x^(1/2)*2^(1/2)+(a/c)^(1/2))/
(x-(a/c)^(1/4)*x^(1/2)*2^(1/2)+(a/c)^(1/2)))-1/2/a^2*c*A*(a/c)^(1/4)*2^(1/2)*arctan(2^(1/2)/(a/c)^(1/4)*x^(1/2
)+1)-1/2/a^2*c*A*(a/c)^(1/4)*2^(1/2)*arctan(2^(1/2)/(a/c)^(1/4)*x^(1/2)-1)-1/4/a*B/(a/c)^(1/4)*2^(1/2)*ln((x-(
a/c)^(1/4)*x^(1/2)*2^(1/2)+(a/c)^(1/2))/(x+(a/c)^(1/4)*x^(1/2)*2^(1/2)+(a/c)^(1/2)))-1/2/a*B/(a/c)^(1/4)*2^(1/
2)*arctan(2^(1/2)/(a/c)^(1/4)*x^(1/2)+1)-1/2/a*B/(a/c)^(1/4)*2^(1/2)*arctan(2^(1/2)/(a/c)^(1/4)*x^(1/2)-1)

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)/x^(5/2)/(c*x^2+a),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [B]  time = 1.5609, size = 1617, normalized size = 5.82 \begin{align*} -\frac{3 \, a x^{2} \sqrt{-\frac{a^{3} \sqrt{-\frac{B^{4} a^{2} c - 2 \, A^{2} B^{2} a c^{2} + A^{4} c^{3}}{a^{7}}} + 2 \, A B c}{a^{3}}} \log \left (-{\left (B^{4} a^{2} c - A^{4} c^{3}\right )} \sqrt{x} +{\left (B a^{6} \sqrt{-\frac{B^{4} a^{2} c - 2 \, A^{2} B^{2} a c^{2} + A^{4} c^{3}}{a^{7}}} - A B^{2} a^{3} c + A^{3} a^{2} c^{2}\right )} \sqrt{-\frac{a^{3} \sqrt{-\frac{B^{4} a^{2} c - 2 \, A^{2} B^{2} a c^{2} + A^{4} c^{3}}{a^{7}}} + 2 \, A B c}{a^{3}}}\right ) - 3 \, a x^{2} \sqrt{-\frac{a^{3} \sqrt{-\frac{B^{4} a^{2} c - 2 \, A^{2} B^{2} a c^{2} + A^{4} c^{3}}{a^{7}}} + 2 \, A B c}{a^{3}}} \log \left (-{\left (B^{4} a^{2} c - A^{4} c^{3}\right )} \sqrt{x} -{\left (B a^{6} \sqrt{-\frac{B^{4} a^{2} c - 2 \, A^{2} B^{2} a c^{2} + A^{4} c^{3}}{a^{7}}} - A B^{2} a^{3} c + A^{3} a^{2} c^{2}\right )} \sqrt{-\frac{a^{3} \sqrt{-\frac{B^{4} a^{2} c - 2 \, A^{2} B^{2} a c^{2} + A^{4} c^{3}}{a^{7}}} + 2 \, A B c}{a^{3}}}\right ) - 3 \, a x^{2} \sqrt{\frac{a^{3} \sqrt{-\frac{B^{4} a^{2} c - 2 \, A^{2} B^{2} a c^{2} + A^{4} c^{3}}{a^{7}}} - 2 \, A B c}{a^{3}}} \log \left (-{\left (B^{4} a^{2} c - A^{4} c^{3}\right )} \sqrt{x} +{\left (B a^{6} \sqrt{-\frac{B^{4} a^{2} c - 2 \, A^{2} B^{2} a c^{2} + A^{4} c^{3}}{a^{7}}} + A B^{2} a^{3} c - A^{3} a^{2} c^{2}\right )} \sqrt{\frac{a^{3} \sqrt{-\frac{B^{4} a^{2} c - 2 \, A^{2} B^{2} a c^{2} + A^{4} c^{3}}{a^{7}}} - 2 \, A B c}{a^{3}}}\right ) + 3 \, a x^{2} \sqrt{\frac{a^{3} \sqrt{-\frac{B^{4} a^{2} c - 2 \, A^{2} B^{2} a c^{2} + A^{4} c^{3}}{a^{7}}} - 2 \, A B c}{a^{3}}} \log \left (-{\left (B^{4} a^{2} c - A^{4} c^{3}\right )} \sqrt{x} -{\left (B a^{6} \sqrt{-\frac{B^{4} a^{2} c - 2 \, A^{2} B^{2} a c^{2} + A^{4} c^{3}}{a^{7}}} + A B^{2} a^{3} c - A^{3} a^{2} c^{2}\right )} \sqrt{\frac{a^{3} \sqrt{-\frac{B^{4} a^{2} c - 2 \, A^{2} B^{2} a c^{2} + A^{4} c^{3}}{a^{7}}} - 2 \, A B c}{a^{3}}}\right ) + 4 \,{\left (3 \, B x + A\right )} \sqrt{x}}{6 \, a x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)/x^(5/2)/(c*x^2+a),x, algorithm="fricas")

[Out]

-1/6*(3*a*x^2*sqrt(-(a^3*sqrt(-(B^4*a^2*c - 2*A^2*B^2*a*c^2 + A^4*c^3)/a^7) + 2*A*B*c)/a^3)*log(-(B^4*a^2*c -
A^4*c^3)*sqrt(x) + (B*a^6*sqrt(-(B^4*a^2*c - 2*A^2*B^2*a*c^2 + A^4*c^3)/a^7) - A*B^2*a^3*c + A^3*a^2*c^2)*sqrt
(-(a^3*sqrt(-(B^4*a^2*c - 2*A^2*B^2*a*c^2 + A^4*c^3)/a^7) + 2*A*B*c)/a^3)) - 3*a*x^2*sqrt(-(a^3*sqrt(-(B^4*a^2
*c - 2*A^2*B^2*a*c^2 + A^4*c^3)/a^7) + 2*A*B*c)/a^3)*log(-(B^4*a^2*c - A^4*c^3)*sqrt(x) - (B*a^6*sqrt(-(B^4*a^
2*c - 2*A^2*B^2*a*c^2 + A^4*c^3)/a^7) - A*B^2*a^3*c + A^3*a^2*c^2)*sqrt(-(a^3*sqrt(-(B^4*a^2*c - 2*A^2*B^2*a*c
^2 + A^4*c^3)/a^7) + 2*A*B*c)/a^3)) - 3*a*x^2*sqrt((a^3*sqrt(-(B^4*a^2*c - 2*A^2*B^2*a*c^2 + A^4*c^3)/a^7) - 2
*A*B*c)/a^3)*log(-(B^4*a^2*c - A^4*c^3)*sqrt(x) + (B*a^6*sqrt(-(B^4*a^2*c - 2*A^2*B^2*a*c^2 + A^4*c^3)/a^7) +
A*B^2*a^3*c - A^3*a^2*c^2)*sqrt((a^3*sqrt(-(B^4*a^2*c - 2*A^2*B^2*a*c^2 + A^4*c^3)/a^7) - 2*A*B*c)/a^3)) + 3*a
*x^2*sqrt((a^3*sqrt(-(B^4*a^2*c - 2*A^2*B^2*a*c^2 + A^4*c^3)/a^7) - 2*A*B*c)/a^3)*log(-(B^4*a^2*c - A^4*c^3)*s
qrt(x) - (B*a^6*sqrt(-(B^4*a^2*c - 2*A^2*B^2*a*c^2 + A^4*c^3)/a^7) + A*B^2*a^3*c - A^3*a^2*c^2)*sqrt((a^3*sqrt
(-(B^4*a^2*c - 2*A^2*B^2*a*c^2 + A^4*c^3)/a^7) - 2*A*B*c)/a^3)) + 4*(3*B*x + A)*sqrt(x))/(a*x^2)

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Sympy [A]  time = 159.829, size = 369, normalized size = 1.33 \begin{align*} A \left (\begin{cases} \frac{\tilde{\infty }}{x^{\frac{7}{2}}} & \text{for}\: a = 0 \wedge c = 0 \\- \frac{2}{7 c x^{\frac{7}{2}}} & \text{for}\: a = 0 \\- \frac{2}{3 a x^{\frac{3}{2}}} & \text{for}\: c = 0 \\- \frac{2}{3 a x^{\frac{3}{2}}} + \frac{\sqrt [4]{-1} \log{\left (- \sqrt [4]{-1} \sqrt [4]{a} \sqrt [4]{\frac{1}{c}} + \sqrt{x} \right )}}{2 a^{\frac{7}{4}} c^{4} \left (\frac{1}{c}\right )^{\frac{19}{4}}} - \frac{\sqrt [4]{-1} \log{\left (\sqrt [4]{-1} \sqrt [4]{a} \sqrt [4]{\frac{1}{c}} + \sqrt{x} \right )}}{2 a^{\frac{7}{4}} c^{4} \left (\frac{1}{c}\right )^{\frac{19}{4}}} + \frac{\sqrt [4]{-1} \operatorname{atan}{\left (\frac{\left (-1\right )^{\frac{3}{4}} \sqrt{x}}{\sqrt [4]{a} \sqrt [4]{\frac{1}{c}}} \right )}}{a^{\frac{7}{4}} c^{4} \left (\frac{1}{c}\right )^{\frac{19}{4}}} & \text{otherwise} \end{cases}\right ) + B \left (\begin{cases} \frac{\tilde{\infty }}{x^{\frac{5}{2}}} & \text{for}\: a = 0 \wedge c = 0 \\- \frac{2}{5 c x^{\frac{5}{2}}} & \text{for}\: a = 0 \\- \frac{2}{a \sqrt{x}} & \text{for}\: c = 0 \\- \frac{2}{a \sqrt{x}} + \frac{\left (-1\right )^{\frac{3}{4}} \log{\left (- \sqrt [4]{-1} \sqrt [4]{a} \sqrt [4]{\frac{1}{c}} + \sqrt{x} \right )}}{2 a^{\frac{5}{4}} c^{2} \left (\frac{1}{c}\right )^{\frac{9}{4}}} - \frac{\left (-1\right )^{\frac{3}{4}} \log{\left (\sqrt [4]{-1} \sqrt [4]{a} \sqrt [4]{\frac{1}{c}} + \sqrt{x} \right )}}{2 a^{\frac{5}{4}} c^{2} \left (\frac{1}{c}\right )^{\frac{9}{4}}} - \frac{\left (-1\right )^{\frac{3}{4}} \operatorname{atan}{\left (\frac{\left (-1\right )^{\frac{3}{4}} \sqrt{x}}{\sqrt [4]{a} \sqrt [4]{\frac{1}{c}}} \right )}}{a^{\frac{5}{4}} c^{2} \left (\frac{1}{c}\right )^{\frac{9}{4}}} & \text{otherwise} \end{cases}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)/x**(5/2)/(c*x**2+a),x)

[Out]

A*Piecewise((zoo/x**(7/2), Eq(a, 0) & Eq(c, 0)), (-2/(7*c*x**(7/2)), Eq(a, 0)), (-2/(3*a*x**(3/2)), Eq(c, 0)),
(-2/(3*a*x**(3/2)) + (-1)**(1/4)*log(-(-1)**(1/4)*a**(1/4)*(1/c)**(1/4) + sqrt(x))/(2*a**(7/4)*c**4*(1/c)**(1
9/4)) - (-1)**(1/4)*log((-1)**(1/4)*a**(1/4)*(1/c)**(1/4) + sqrt(x))/(2*a**(7/4)*c**4*(1/c)**(19/4)) + (-1)**(
1/4)*atan((-1)**(3/4)*sqrt(x)/(a**(1/4)*(1/c)**(1/4)))/(a**(7/4)*c**4*(1/c)**(19/4)), True)) + B*Piecewise((zo
o/x**(5/2), Eq(a, 0) & Eq(c, 0)), (-2/(5*c*x**(5/2)), Eq(a, 0)), (-2/(a*sqrt(x)), Eq(c, 0)), (-2/(a*sqrt(x)) +
(-1)**(3/4)*log(-(-1)**(1/4)*a**(1/4)*(1/c)**(1/4) + sqrt(x))/(2*a**(5/4)*c**2*(1/c)**(9/4)) - (-1)**(3/4)*lo
g((-1)**(1/4)*a**(1/4)*(1/c)**(1/4) + sqrt(x))/(2*a**(5/4)*c**2*(1/c)**(9/4)) - (-1)**(3/4)*atan((-1)**(3/4)*s
qrt(x)/(a**(1/4)*(1/c)**(1/4)))/(a**(5/4)*c**2*(1/c)**(9/4)), True))

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Giac [A]  time = 1.19785, size = 348, normalized size = 1.25 \begin{align*} -\frac{2 \,{\left (3 \, B x + A\right )}}{3 \, a x^{\frac{3}{2}}} - \frac{\sqrt{2}{\left (\left (a c^{3}\right )^{\frac{1}{4}} A c^{2} + \left (a c^{3}\right )^{\frac{3}{4}} B\right )} \arctan \left (\frac{\sqrt{2}{\left (\sqrt{2} \left (\frac{a}{c}\right )^{\frac{1}{4}} + 2 \, \sqrt{x}\right )}}{2 \, \left (\frac{a}{c}\right )^{\frac{1}{4}}}\right )}{2 \, a^{2} c^{2}} - \frac{\sqrt{2}{\left (\left (a c^{3}\right )^{\frac{1}{4}} A c^{2} + \left (a c^{3}\right )^{\frac{3}{4}} B\right )} \arctan \left (-\frac{\sqrt{2}{\left (\sqrt{2} \left (\frac{a}{c}\right )^{\frac{1}{4}} - 2 \, \sqrt{x}\right )}}{2 \, \left (\frac{a}{c}\right )^{\frac{1}{4}}}\right )}{2 \, a^{2} c^{2}} - \frac{\sqrt{2}{\left (\left (a c^{3}\right )^{\frac{1}{4}} A c^{2} - \left (a c^{3}\right )^{\frac{3}{4}} B\right )} \log \left (\sqrt{2} \sqrt{x} \left (\frac{a}{c}\right )^{\frac{1}{4}} + x + \sqrt{\frac{a}{c}}\right )}{4 \, a^{2} c^{2}} + \frac{\sqrt{2}{\left (\left (a c^{3}\right )^{\frac{1}{4}} A c^{2} - \left (a c^{3}\right )^{\frac{3}{4}} B\right )} \log \left (-\sqrt{2} \sqrt{x} \left (\frac{a}{c}\right )^{\frac{1}{4}} + x + \sqrt{\frac{a}{c}}\right )}{4 \, a^{2} c^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)/x^(5/2)/(c*x^2+a),x, algorithm="giac")

[Out]

-2/3*(3*B*x + A)/(a*x^(3/2)) - 1/2*sqrt(2)*((a*c^3)^(1/4)*A*c^2 + (a*c^3)^(3/4)*B)*arctan(1/2*sqrt(2)*(sqrt(2)
*(a/c)^(1/4) + 2*sqrt(x))/(a/c)^(1/4))/(a^2*c^2) - 1/2*sqrt(2)*((a*c^3)^(1/4)*A*c^2 + (a*c^3)^(3/4)*B)*arctan(
-1/2*sqrt(2)*(sqrt(2)*(a/c)^(1/4) - 2*sqrt(x))/(a/c)^(1/4))/(a^2*c^2) - 1/4*sqrt(2)*((a*c^3)^(1/4)*A*c^2 - (a*
c^3)^(3/4)*B)*log(sqrt(2)*sqrt(x)*(a/c)^(1/4) + x + sqrt(a/c))/(a^2*c^2) + 1/4*sqrt(2)*((a*c^3)^(1/4)*A*c^2 -
(a*c^3)^(3/4)*B)*log(-sqrt(2)*sqrt(x)*(a/c)^(1/4) + x + sqrt(a/c))/(a^2*c^2)